A finite difference method for an initial–boundary value problem with a Riemann–Liouville–Caputo spatial fractional derivative
José Luis Gracia,Martin Stynes +1 more
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TL;DR: A fractional Friedrichs’ inequality is derived and is used to prove that the problem approaches a steady-state solution when the source term is zero, and it is proved that the scheme converges with first order in the maximum norm.
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About: This article is published in Journal of Computational and Applied Mathematics. The article was published on 01 Jan 2021. and is currently open access. The article focuses on the topics: Fractional calculus & Boundary value problem.
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Figures
Figure 3: Example 3: Computed solutions with the method (12) for N = M = 64 and α = 1.2, 1.4, 1.6, 1.8. 
Figure 4: Example 3: Approximated values of ‖u(·, tn)‖L2(0,1) using (12) with N = M = 256 and α = 1.2 (solid line), α = 1.4 (dashed line), α = 1.6 (dotted line) and α = 1.8 (dash-dotted line). 
Table 3: Example 2: Maximum two-mesh nodal differences and orders of convergence 
Table 4: Example 2: Maximum two-mesh nodal differences and orders of convergence 
Figure 1: Example 2: Computed solutions with the method (12) for N = M = 64 and α = 1.2, 1.6. 
Table 2: Example 1: Maximum nodal errors and orders of convergence
Citations
•Posted Content
Boundary Conditions for Fractional Diffusion
TL;DR: In this paper, the authors derived physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach, and theoretical properties, including well-posedness and steady state solutions, were reviewed.
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References
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Analysis of Fractional Differential Equations
Kai Diethelm,Neville J. Ford +1 more
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The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type
Kai Diethelm
- 02 Dec 2010
TL;DR: In this paper, the existence and uniqueness results for Riemann-Liouville Fractional Differential Equations are presented. But they do not cover the special cases of fractional calculus.
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Robust Computational Techniques for Boundary Layers
Paul A. Farrell,Alan F. Hegarty,John M. Miller,Eugene O'Riordan,G. I. Shishkin +4 more
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TL;DR: In this paper, numerical methods for problems with Boundary Layers are presented. But they do not address the problems with Frictionless Walls and No Slip Boundary Conditions, and they are not suitable for Non-Monotone Methods in two dimensions.
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